Exploring the solution space of sorting by reversals, with experiments and an application to evolution

In comparative genomics, algorithms that sort permutations by reversals are often used to propose evolutionary scenarios of rearrangements between species. One of the main problems of such methods is that they give one solution while the number of optimal solutions is huge, with no criteria to discriminate among them. Bergeron et al. started to give some structure to the set of optimal solutions, in order to be able to deliver more presentable results than only one solution or a complete list of all solutions. However, no algorithm exists so far to compute this structure except through the enumeration of all solutions, which takes too much time even for small permutations. Bergeron et al. state as an open problem the design of such an algorithm. We propose in this paper an answer to this problem, that is, an algorithm which gives all the classes of solutions and counts the number of solutions in each class, with a better theoretical and practical complexity than the complete enumeration method. We give an example of how to reduce the number of classes obtained, using further constraints. Finally, we apply our algorithm to analyse the possible scenarios of rearrangement between mammalian sex chromosomes.     

An algorithm to enumerate sorting reversals for signed permutations.

The rearrangement distance between single-chromosome genomes can be estimated as the minimum number of inversions required to transform the gene ordering observed in one into that observed in the other. This measure, known as "inversion distance," can be computed as the reversal distance between signed permutations. During the past decade, much progress has been made both on the problem of computing reversal distance and on the related problem of finding a minimum-length sequence of reversals, which is known as "sorting by reversals." For most problem instances, however, many minimum-length sequences of reversals exist, and in the absence of auxiliary information, no one is of greater value than the others. The problem of finding all minimum-length sequences of reversals is thus a natural generalization of sorting by reversals, yet it has received little attention. This problem reduces easily to the problem of finding all "sorting reversals" of one permutation with respect to another - that is, all reversals rho such that, if rho is applied to one permutation, then the reversal distance of that permutation from the other is decreased. In this paper, an efficient algorithm is derived to solve the problem of finding all sorting reversals, and experimental results are presented indicating that, while the new algorithm does not represent a significant improvement in asymptotic terms (it takes O(n(3)) time, for permutations of size n; the problem can now be solved by brute force in Theta(n(3)) time), it performs dramatically better in practice than the best known alternative. An implementation of the algorithm is available at www.cse.ucsc.edu/~acs.     

Sorting by restricted-length-weighted reversals

Classical sorting by reversals uses the unit-cost model, that is, each reversal consumes an equal cost. This model limits the biological meaning of sorting by reversal. Bender and his colleagues extended it by assigning a cost function f（1） = l^a for all a≥ 0, where l is the length of the reversed subsequence. In this paper, we extend their results by considering a model in which long reversals are prohibited. Using the same cost function above for permitted reversals, we present tight or nearly tight bounds for the worst-case cost of sorting by reversals. Then we develop algorithms to approximate the optimal cost to sort a given 0/1 sequence as well as a given permutation. Our proposed problems are more biologically meaningful and more algorithmically general and challenging than the problem considered by Bender et al. Furthermore, our bounds are tight and nearly tight, whereas our algorithms provide good approximation ratios compared to the optimal cost to sort 0/1 sequences or permutations by reversals.     

Two notes on genome rearrangement

A central problem in genome rearrangement is finding a most parsimonious rearrangement scenario using certain rearrangement operations. An important problem of this type is sorting a signed genome by reversals and translocations (SBRT). Hannenhalli and Pevzner presented a duality theorem for SBRT which leads to a polynomial time algorithm for sorting a multi-chromosomal genome using a minimum number of reversals and translocations. However, there is one case for which their theorem and algorithm fail. We describe that case and suggest a correction to the theorem and the polynomial algorithm. The solution of SBRT uses a reduction to the problem of sorting a signed permutation by reversals (SBR). The best extant algorithms for SBR require quadratic time. The common approach to solve SBR is by finding a safe reversal using the overlap graph or the interleaving graph of a permutation. We describe a family of signed permutations which proves a quadratic lower bound on the number of affected vertices in the overlap/interleaving graph during any optimal sorting scenario. This implies, in particular, an Omega(n3) lower bound for Bergeron's algorithm.     

Perfect sorting by reversals is not always difficult

We propose new algorithms for computing pairwise rearrangement scenarios that conserve the combinatorial structure of genomes. More precisely, we investigate the problem of sorting signed permutations by reversals without breaking common intervals. We describe a combinatorial framework for this problem that allows us to characterize classes of signed permutations for which one can compute, in polynomial time, a shortest reversal scenario that conserves all common intervals. In particular, we define a class of permutations for which this computation can be done in linear time with a very simple algorithm that does not rely on the classical Hannenhalli-Pevzner theory for sorting by reversals. We apply these methods to the computation of rearrangement scenarios between permutations obtained from 16 synteny blocks of the X chromosomes of the human, mouse, and rat     

Genome rearrangement based on reversals that preserve conserved intervals

The order of genes in the genomes of species can change during evolution and can provide information about their phylogenetic relationship. An interesting method to infer the phylogenetic relationship from the gene orders is to use different types of rearrangement operations and to find possible rearrangement scenarios using these operations. One of the most common rearrangement operations is reversals, which reverse the order of a subset of neighbored genes. In this paper, we study the problem to find the ancestral gene order for three species represented by their gene orders. The rearrangement scenario should use a minimal number of reversals and no other rearrangement operations. This problem is called the Median problem and is known to be NP--complete. In this paper, we describe a heuristic algorithm for finding solutions to the Median problem that searches for rearrangement scenarios with the additional property that gene groups should not be destroyed by reversal operations. The concept of conserved intervals for signed permutations is used to describe such gene groups. We show experimentally, for different types of test problems, that the proposed algorithm produces very good results compared to other algorithms for the Median problem. We also integrate our reversal selection procedure into the well-known MGR and GRAPPA algorithms and show that they achieve a significant speedup while obtaining solutions of the same quality as the original algorithms on the test problems.     

Evolution under reversals: parsimony and conservation of common intervals

In comparative genomics, gene order data is often modeled as signed permutations. A classical problem for genome comparison is to detect common intervals in permutations, that is, genes that are colocalized in several species, indicating that they remained grouped during evolution. A second largely studied problem related to gene order is to compute a minimum scenario of reversals that transforms a signed permutation into another. Several studies began to mix the two problems and it was observed that their results are not always compatible: Often, parsimonious scenarios of reversals break common intervals. If a scenario does not break any common interval, it is called perfect. In two recent studies, Berard et al. defined a class of permutations for which building a perfect scenario of reversals sorting a permutation was achieved in polynomial time and stated as an open question whether it is possible to decide, given a permutation, if there exists a minimum scenario of reversals that is perfect. In this paper, we give a solution to this problem and prove that this widens the class of permutations addressed by the aforementioned studies. We implemented and tested this algorithm on gene order data of chromosomes from several mammal species and we compared it to other methods. The algorithm helps to choose among several possible scenarios of reversals and indicates that the minimum scenario of reversals is not always the most plausible     

New bounds and tractable instances for the transposition distance

The problem of sorting by transpositions asks for a sequence of adjacent interval exchanges that sorts a permutation and is of the shortest possible length. The distance of the permutation is defined as the length of such a sequence. Despite the apparently intuitive nature of this problem, introduced in 1995 by Bafna and Pevzner, the complexity of both finding an optimal sequence and computing the distance remains open today. In this paper, we establish connections between two different graph representations of permutations, which allows us to compute the distance of a few nontrivial classes of permutations in linear time and space, bypassing the use of any graph structure. By showing that every permutation can be obtained from one of these classes, we prove a new tight upper bound on the transposition distance. Finally, we give improved bounds on some other families of permutations and prove formulas for computing the exact distance of other classes of permutations, again in polynomial time     

Common intervals and sorting by reversals: a marriage of necessity

This paper revisits the problem of sorting by reversals with tools developed in the context of detecting common intervals. Mixing the two approaches yields new definitions and algorithms for the reversal distance computations, that apply directly on the original permutation. Traditional constructions such as recasting the signed permutation as a positive permutation, or traversing the overlap graph to analyze its connected components, are replaced by elementary definitions in terms of intervals of the permutation. This yields simple linear time algorithms that identify the essential features in a single pass over the permutation and use only simple data structures like arrays and stacks.     

Atomic-Accuracy Prediction of Protein Loop Structures through an RNA-Inspired Ansatz

Consistently predicting biopolymer structure at atomic resolution from sequence alone remains a difficult problem, even for small sub-segments of large proteins. Such loop prediction challenges, which arise frequently in comparative modeling and protein design, can become intractable as loop lengths exceed 10 residues and if surrounding side-chain conformations are erased. Current approaches, such as the protein local optimization protocol or kinematic inversion closure (KIC) Monte Carlo, involve stages that coarse-grain proteins, simplifying modeling but precluding a systematic search of all-atom configurations. This article introduces an alternative modeling strategy based on a ‘stepwise ansatz’, recently developed for RNA modeling, which posits that any realistic all-atom molecular conformation can be built up by residue-by-residue stepwise enumeration. When harnessed to a dynamic-programming-like recursion in the Rosetta framework, the resulting stepwise assembly (SWA) protocol enables enumerative sampling of a 12 residue loop at a significant but achievable cost of thousands of CPU-hours. In a previously established benchmark, SWA recovers crystallographic conformations with sub-Angstrom accuracy for 19 of 20 loops, compared to 14 of 20 by KIC modeling with a comparable expenditure of computational power. Furthermore, SWA gives high accuracy results on an additional set of 15 loops highlighted in the biological literature for their irregularity or unusual length. Successes include cis-Pro touch turns, loops that pass through tunnels of other side-chains, and loops of lengths up to 24 residues. Remaining problem cases are traced to inaccuracies in the Rosetta all-atom energy function. In five additional blind tests, SWA achieves sub-Angstrom accuracy models, including the first such success in a protein/RNA binding interface, the YbxF/kink-turn interaction in the fourth ‘RNA-puzzle’ competition. These results establish all-atom enumeration as an unusually systematic approach to ab initio protein structure modeling that can leverage high performance computing and physically realistic energy functions to more consistently achieve atomic accuracy.     

The top-scoring 'N' algorithm: a generalized relative expression classification method from small numbers of biomolecules

ABSTRACT: BACKGROUND: Relative expression algorithms such as the top-scoring pair (TSP) and the top-scoring triplet (TST) have several strengths that distinguish them from other classification methods, including resistance to overfitting, invariance to most data normalization methods, and biological interpretability. The top-scoring 'N' (TSN) algorithm is a generalized form of other relative expression algorithms which uses generic permutations and a dynamic classifier size to control both the permutation and combination space available for classification. RESULTS: TSN was tested on nine cancer datasets, showing statistically significant differences in classification accuracy between different classifier sizes (choices of N). TSN also performed competitively against a wide variety of different classification methods, including artificial neural networks, classification trees, discriminant analysis, k-Nearest neighbor, naive Bayes, and support vector machines, when tested on the Microarray Quality Control II datasets. Furthermore, TSN exhibits low levels of overfitting on training data compared to other methods, giving confidence that results obtained during cross validation will be more generally applicable to external validation sets. CONCLUSIONS: TSN preserves the strengths of other relative expression algorithms while allowing a much larger permutation and combination space to be explored, potentially improving classification accuracies when fewer numbers of measured features are available.     

Sorting signed circular permutations by super short operations

Background

One way to estimate the evolutionary distance between two given genomes is to determine the minimum number of large-scale mutations, or genome rearrangements, that are necessary to transform one into the other. In this context, genomes can be represented as ordered sequences of genes, each gene being represented by a signed integer. If no gene is repeated, genomes are thus modeled as signed permutations of the form \(\pi =(\pi _1 \pi _2 \ldots \pi _n)\), and in that case we can consider without loss of generality that one of them is the identity permutation \(\iota _n =(1 2 \ldots n)\), and that we just need to sort the other (i.e., transform it into \(\iota _n\)). The most studied genome rearrangement events are reversals, where a segment of the genome is reversed and reincorporated at the same location; and transpositions, where two consecutive segments are exchanged. Many variants, e.g., combining different types of (possibly constrained) rearrangements, have been proposed in the literature. One of them considers that the number of genes involved, in a reversal or a transposition, is never greater than two, which is known as the problem of sorting by super short operations (or SSOs).

Results and conclusions

All problems considering SSOs in permutations have been shown to be in \(\mathsf {P}\), except for one, namely sorting signed circular permutations by super short reversals and super short transpositions. Here we fill this gap by introducing a new graph structure called cyclic permutation graph and providing a series of intermediate results, which allows us to design a polynomial algorithm for sorting signed circular permutations by super short reversals and super short transpositions.

     

A greedier approach for finding tag SNPs

MOTIVATION: Recent studies have shown that a small subset of Single Nucleotide Polymorphisms (SNPs) (called tag SNPs) is sufficient to capture the haplotype patterns in a high linkage disequilibrium region. To find the minimum set of tag SNPs, exact algorithms for finding the optimal solution could take exponential time. On the other hand, approximation algorithms are more efficient but may fail to find the optimal solution. RESULTS: We propose a hybrid method that combines the ideas of the branch-and-bound method and the greedy algorithm. This method explores larger solution space to obtain a better solution than a traditional greedy algorithm. It also allows the user to adjust the efficiency of the program and quality of solutions. This algorithm has been implemented and tested on a variety of simulated and biological data. The experimental results indicate that our program can find better solutions than previous methods. This approach is quite general since it can be used to adapt other greedy algorithms to solve their corresponding problems. AVAILABILITY: The program is available upon request.     

Sorting by short swaps.

A short swap is an operation on a permutation that switches two elements that have at most one element between them. This paper investigates the problem of finding a minimum-length sorting sequence of short swaps for a given permutation. A polynomial-time 2-approximation algorithm for this problem is presented, and bounds for the short-swap diameter (the length of the longest minimum sorting sequence among all permutations of a given length) are also obtained.     

Performance assessment of protein multiple sequence alignment algorithms based on permutation similarity measurement

Protein multiple sequence alignment is an important bioinformatics tool. It has important applications in biological evolution analysis and protein structure prediction. A variety of alignment algorithms in this field have achieved great success. However, each algorithm has its own inherent deficiencies. In this paper, permutation similarity is proposed to evaluate several protein multiple sequence alignment algorithms that are widely used currently. As the permutation similarity method only concerns the relative order of different protein evolutionary distances, without taking into account the slight difference between the evolutionary distances, it can get more robust evaluations. The longest common subsequence method is adopted to define the similarity between different permutations. Using these methods, we assessed Dialign, Tcoffee, ClustalW and Muscle and made comparisons among them.     

Online clustering algorithms

We introduce a set of clustering algorithms whose performance function is such that the algorithms overcome one of the weaknesses of K-means, its sensitivity to initial conditions which leads it to converge to a local optimum rather than the global optimum. We derive online learning algorithms and illustrate their convergence to optimal solutions which K-means fails to find. We then extend the algorithm by underpinning it with a latent space which enables a topology preserving mapping to be found. We show visualisation results on some standard data sets.     

Naturally occurring circular permutations in proteins

A pair of proteins is defined to be related by a circular permutation if the N-terminal region of one protein has significant sequence similarity to the C-terminal of the other and vice versa. To detect pairs of proteins that might be related by circular permutation, we implemented a procedure based on a combination of a fast screening algorithm that we had designed and manual verification of candidate pairs. The screening algorithm is a variation of a dynamic programming string matching algorithm, in which one of the sequences is doubled. This algorithm, although not guaranteed to identify all cases of circular permutation, is a good first indicator of protein pairs related by permutation events. The candidate pairs were further validated first by application of an exhaustive string matching algorithm and then by manual inspection using the dotplot visual tool. Screening the whole Swissprot database, a total of 25 independent protein pairs were identified. These cases are presented here, divided into three categories depending on the level of functional similarity of the related proteins. To validate our approach and to confirm further the small number of circularly permuted protein pairs, a systematic search for cases of circular permutation was carried out in the Pfam database of protein domains. Even with this more inclusive definition of a circular permutation, only seven additional candidates were found. None of these fitted our original definition of circular permutations. The small number of cases of circular permutation suggests that there is no mechanism of local genetic manipulation that can induce circular permutations; most examples observed seem to result from fusion of functional units.     

Fuzzy K-nearest neighbor classifiers for ventricular arrhythmia detection

We report a study of the efficiency of 4 classifiers (the K-nearest-neighbor and single-nearest-prototype algorithms, each as parametrized by both Fuzzy C-Means and Fuzzy Covariance clustering) in the detection of ventricular arrhythmias in ECG traces characterized by 4 features derived from 7 spectral parameters. Principal components analysis was used in conjunction with a cardiologist's deterministic classification of 90 ECG traces to fix the number of trace classes to 5 (ventricular fibrillation/flutter, sinus rhythm, ventricular rhythms with aberrant complexes and 2 classes of artefact). Forty of the 90 traces were then defined as a test set; 5 different learning sets (numbering 25, 30, 35, 40 and 45 traces) were randomly selected from the remaining 50 traces; each learning set was used to parametrize both the classification algorithms using both fuzzy clustering algorithms and the parametrized classification algorithms were then applied to the test set. Optimal K for K-nearest-neighbor algorithms and optimal cluster volumes for Fuzzy Covariance algorithms were sought by trial and error to minimize classification differences with respect to the cardiologist's classification. Fuzzy Covariance clustering afforded significantly better perception of cluster structure than the Fuzzy C-Means algorithm, and the classifiers performed correspondingly with an overall empirical error ratio of just 0.10 for the K-nearest-neighbor algorithm parametrized by Fuzzy Covariance.     

A lower bound on the reversal and transposition diameter.

One possible model to study genome evolution is to represent genomes as permutations of genes and compute distances based on the minimum number of certain operations (rearrangements) needed to transform one permutation into another. Under this model, the shorter the distance, the closer the genomes are. Two operations that have been extensively studied are the reversal and the transposition. A reversal is an operation that reverses the order of the genes on a certain portion of the permutation. A transposition is an operation that "cuts" a certain portion of the permutation and "pastes" it elsewhere in the same permutation. In this note, we show that the reversal and transposition distance of the signed permutation pi(n) = (-1 -2.-(n - 1)-n) with respect to the identity is left floor n/2 right floor + 2 for all n>or=3. We conjecture that this value is the diameter of the permutation group under these operations.